I wish to calculate (or upper bound) expectations of the form $E[\langle x,y \rangle^2]$, where $x$ and $y$ are i.i.d standard gaussian vectors of length n. Are there any exponential type upper bounds for the same?

I think I am misunderstanding the question along the same lines as Igor Rivin's answer below... another way of phrasing his answer is: let $W_i = x_iy_i$, then since each $W_i$ has mean zero you are looking at the variance of a sum of $n$ i.i.d. copies of $W_1$, and $W_1$ has variance $1$, so the whole sum has variance $n$.
–
Yemon ChoiNov 26 '11 at 0:05

1 Answer
1

I must be misunderstanding the question, but $<x, y>^2$ is a sum of the terms of the form $x_i x_j y_i y_j.$ The expectation of this term vanishes, unless $i=j,$ in which case it (the . expectation) is the square of the expectation of the square of the standard Gaussian. The square of the standard gaussian is the chi-square distribution with one degree of freedom, whose mean is $1,$ so the whole thing should be $n$.

Perhaps I didnt pose the question correctly. What I am looking for is something along these lines: let $\theta = <x,y>$, where x and y are as defined earlier. so $\theta$ is a sum of bessel functions, whose mean is 0. I am now looking to upper bound the expected value of $\theta^2$ .
–
user19530Nov 25 '11 at 22:35

2

Perhaps you can explain Igor's misunderstanding, since I have it too.
–
Brendan McKayNov 26 '11 at 14:59